Chapter XII: Act 1890: , the effect of which is explained in the article Insanity. Any (8)

Multiple Integrals.

53. The meaning of integration of a function of n variables through a
domain of the same number of dimensions is explained in the article
FUNCTION. In the case of two variables x, y we integrate a function
[f](x, y) over an area; in the case of three variables x, y, z we
integrate a function [f](x, y, z) through a volume. The integral of a
function [f](x, y) over an area in the plane of (x, y) is denoted by
_ _
/ /
| | [f](x, y) dx dy.
_/_/

The notation refers to a method of evaluating the integral. We may
suppose the area divided into a very large number of very small
rectangles by lines parallel to the axes. Then we multiply the value
of [f] at any point within a rectangle by the measure of the area of
the rectangle, sum for all the rectangles, and pass to a limit by
increasing the number of rectangles indefinitely and diminishing all
their sides indefinitely. The process is usually effected by summing
first for all the rectangles which lie in a strip between two lines
parallel to one axis, say the axis of y, and afterwards for all the
strips. This process is equivalent to integrating [f](x, y) with
respect to y, keeping x constant, and taking certain functions of x as
the limits of integration for y, and then integrating the result with
respect to x between constant limits. The integral obtained in this
way may be written in such a form as
_ _
/ b { / [f]2(x) }
| dx { | [f](x, y) dy },
_/ a { _/ [f]1(x) }

and is called a "repeated integral." The identification of a surface
integral, such as [int][int][f](x, y)dxdy, with a repeated integral
cannot always be made, but implies that the function satisfies certain
conditions of continuity. In the same way volume integrals are usually
evaluated by regarding them as repeated integrals, and a volume
integral is written in the form
_ _ _
/ / /
| | | [f](x, y, z) dx dy dz.
_/_/_/

Integrals such as surface and volume integrals are usually called
"multiple integrals." Thus we have "double" integrals, "triple"
integrals, and so on. In contradistinction to multiple integrals the
ordinary integral of a function of one variable with respect to that
variable is called a "simple integral."

Surface Integrals.

A more general type of surface integral may be defined by taking an
arbitrary surface, with or without an edge. We suppose in the first
place that the surface is closed, or has no edge. We may mark a large
number of points on the surface, and draw the tangent planes at all
these points. These tangent planes form a polyhedron having a large
number of faces, one to each marked point; and we may choose the
marked points so that all the linear dimensions of any face are less
than some arbitrarily chosen length. We may devise a rule for
increasing the number of marked points indefinitely and decreasing the
lengths of all the edges of the polyhedra indefinitely. If the sum of
the areas of the faces tends to a limit, this limit is the area of the
surface. If we multiply the value of a function [f] at a point of the
surface by the measure of the area of the corresponding face of the
polyhedron, sum for all the faces, and pass to a limit as before, the
result is a surface integral, and is written
_ _
/ /
| | [f] dS.
_/_/

Line Integrals.

The extension to the case of an open surface bounded by an edge
presents no difficulty. A line integral taken along a curve is defined
in a similar way, and is written
_
/
| [f] ds
_/

where ds is the element of arc of the curve (§ 33). The direction
cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line
integrals usually present themselves in the form
_ _
/ / dx dy dz \ /
| ( u -- + v -- + w -- ) ds or | (u dx + v dy + w dz).
_/ \ ds ds ds / _/ s

In like manner surface integrals usually present themselves in the
form
_ _
/ /
| | (l[xi] + m[eta] + n[zeta]) dS
_/_/

where l, m, n are the direction cosines of the normal to the surface
drawn in a specified sense.

The area of a bounded portion of the plane of (x, y) may be expressed
either as
_
/
½ | (x dy - y dx),
_/

or as
_ _
/ /
| | dx dy,
_/_/

the former integral being a line integral taken round the boundary of
the portion, and the latter a surface integral taken over the area
within this boundary. In forming the line integral the boundary is
supposed to be described in the positive sense, so that the included
area is on the left hand.

Theorems of Green and Stokes.

53_a_. We have two theorems of transformation connecting volume
integrals with surface integrals and surface integrals with line
integrals. The first theorem, called "Green's theorem," is expressed
by the equation
_ _ _ _ _
/ / / / ð[xi] ð[eta] ð[zeta]\ / /
| | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS,
_/_/_/ \ ðx ðy ðz / _/_/

where the volume integral on the left is taken through the volume
within a closed surface S, and the surface integral on the right is
taken over S, and l, m, n denote the direction cosines of the normal
to S drawn outwards. There is a corresponding theorem for a closed
curve in two dimensions, viz.,
_ _ _
/ / / ð[xi] ð[eta]\ / / dy dx \
| | ( ----- + ------ ) dx dy = | ( [xi] -- - [eta] -- ) ds,
_/_/ \ ðx ðy / _/ \ ds ds /

the sense of description of s being the positive sense. This theorem
is a particular case of a more general theorem called "Stokes's
theorem." Let s denote the edge of an open surface S, and let S be
covered with a network of curves so that the meshes of the network are
nearly plane, then we can choose a sense of description of the edge of
any mesh, and a corresponding sense for the normal to S at any point
within the mesh, so that these senses are related like the directions
of rotation and translation in a right-handed screw. This convention
fixes the sense of the normal (l, m, n) at any point on S when the
sense of description of s is chosen. If the axes of x, y, z are a
right-handed system, we have Stokes's theorem in the form
_ _ _
/ / / { /ðw ðv\ /ðu ðw\ /ðv ðu\ }
| (u dx + v dy + w dz) = | | { l( -- - -- ) + m( -- - -- ) + n( -- - -- ) }dS,
_/ s _/_/ { \ðy ðz/ \ðz ðx/ \ðx ðy/ }

where the integral on the left is taken round the curve s in the
chosen sense. When the axes are left-handed, we may either reverse the
sense of l, m, n and maintain the formula, or retain the sense of l,
m, n and change the sign of the right-hand member of the equation. For
the validity of the theorems of Green and Stokes it is in general
necessary that the functions involved should satisfy certain
conditions of continuity. For example, in Green's theorem the
differential coefficients ð[xi]/ðx, ð[eta]/ðy, ð[zeta]/ðz must be
continuous within S. Further, there are restrictions upon the nature
of the curves or surfaces involved. For example, Green's theorem, as
here stated, applies only to simply-connected regions of space. The
correction for multiply-connected regions is important in several
physical theories.

Change of Variables in a Multiple Integral.

54. The process of changing the variables in a multiple integral, such
as a surface or volume integral, is divisible into two stages. It is
necessary in the first place to determine the differential element
expressed by the product of the differentials of the first set of
variables in terms of the differentials of the second set of
variables. It is necessary in the second place to determine the limits
of integration which must be employed when the integral in terms of
the new variables is evaluated as a repeated integral. The first part
of the problem is solved at once by the introduction of the Jacobian.
If the variables of one set are denoted by x1, x2, ..., x_n, and those
of the other set by u1, u2, ..., u_n, we have the relation

ð(x1, x2, ..., x_n)
dx1 dx2 ...dx_n = ------------------- du1 du2 ... du_n.
ð(u1, u2, ..., u_n)

In regard to the second stage of the process the limits of integration
must be determined by the rule that the integration with respect to
the second set of variables is to be taken through the same domain as
the integration with respect to the first set.

For example, when we have to integrate a function [f](x, y) over the
area within a circle given by x² + y² = a², and we introduce polar
coordinates so that x = r cos [theta], y = r sin [theta], we find that
r is the value of the Jacobian, and that all points within or on the
circle are given by a [>=] r [>=] o, 2[pi][>=][theta][>=]o, and we have
_ _ _ _
/ a / [root](a²-x²) / a /2[pi]
| dx | [f](x, y) dy = | dr | f(r cos [theta], r sin [theta]) r d[theta].
_/-a _/-[root](a²-x²) _/ 0 _/ 0

If we have to integrate over the area of a rectangle a [>=] x [>=] 0,
b [>=] y [>=] 0, and we transform to polar coordinates, the integral
becomes the sum of two integrals, as follows:--
_ _ _ _
/a / b /tan^-1 b/a /a sec [theta]
| dx | [f](x, y) dy = | d[theta] | [f](r cos [theta], r sin [theta]) r dr
_/0 _/ 0 _/ 0 _/0
_ _
/ ½[pi] /b cosec [theta]
+ | d[theta] | [f](r cos [theta], r sin [theta]) r dr.
_/tan^-1 b/a _/ 0

55. A few additional results in relation to line integrals and
multiple integrals are set down here.

Line Integrals and Multiple Integrals.

(i.) Any simple integral can be regarded as a line-integral taken
along a portion of the axis of x. When a change of variables is made,
the limits of integration with respect to the new variable must be
such that the domain of integration is the same as before. This
condition may require the replacing of the original integral by the
sum of two or more simple integrals.

(ii.) The line integral of a perfect differential of a one-valued
function, taken along any closed curve, is zero.

(iii.) The area within any plane closed curve can be expressed by
either of the formulae
_ _
/ /
| ½ r² d[theta] or | ½ p ds,
_/ _/

where r, [theta] are polar coordinates, and p is the perpendicular
drawn from a fixed point to the tangent. The integrals are to be
understood as line integrals taken along the curve. When the same
integrals are taken between limits which correspond to two points of
the curve, in the sense of line integrals along the arc between the
points, they represent the area bounded by the arc and the terminal
radii vectores.

(iv.) The volume enclosed by a surface which is generated by the
revolution of a curve about the axis of x is expressed by the formula
_
/
[pi] | y² dx,
_/

and the area of the surface is expressed by the formula
_
/
2[pi] | y ds,
_/

where ds is the differential element of arc of the curve. When the
former integral is taken between assigned limits it represents the
volume contained between the surface and two planes which cut the axis
of x at right angles. The latter integral is to be understood as a
line integral taken along the curve, and it represents the area of the
portion of the curved surface which is contained between two planes at
right angles to the axis of x.

(v.) When we use curvilinear coordinates [xi], [eta] which are
conjugate functions of x, y, that is to say are such that

ð[xi]/ðx = ð[eta]/ðy and ð[xi]/ðy = -ð[eta]/ðx,

the Jacobian ð([xi], [eta])/ð(x, v) can be expressed in the form

/ð[xi]\² /ð[eta]\²
( ----- ) + ( ------ ),
\ ðx / \ ðx /

and in a number of equivalent forms. The area of any portion of the
plane is represented by the double integral
_ _
/ /
| | J^-1 d[xi] d[eta],
_/_/

where J denotes the above Jacobian, and the integration is taken
through a suitable domain. When the boundary consists of portions of
curves for which [xi] = const., or [eta] = const., the above is
generally the simplest way of evaluating it.

(vi.) The problem of "rectifying" a plane curve, or finding its
length, is solved by evaluating the integral
_
/ { /dy\² }½
| { 1 + ( -- ) } dx,
_/ { \dx/ }

or, in polar coordinates, by evaluating the integral
_
/ { / dr \² }½
| { r² + ( -------- ) } d[theta].
_/ { \d[theta]/ }

In both cases the integrals are line integrals taken along the curve.

(vii.) When we use curvilinear coordinates [xi], [eta] as in (v.)
above, the length of any portion of a curve [xi] = const. is given by
the integral
_
/
| J^-½ d[eta]
_/

taken between appropriate limits for [eta]. There is a similar formula
for the arc of a curve [eta] = const.

(viii.) The area of a surface z = [f](x, y) can be expressed by the
formula
_ _
/ / { /ðz\² /ðz\² }½
| | { 1 + ( -- ) + ( -- ) } dx dy.
_/_/ { \ðx/ \ðy/ }

When the coordinates of the points of a surface are expressed as
functions of two parameters u, v, the area is expressed by the
formula
_ _ _ _
/ / | { ð(y, z) }² { ð(z, x) }² { ð(x, y) }² |½
| | | { ------- } + { ------- } + { ------- } | du dv.
_/_/ |_ { ð(u, v) } { ð(u, v) } { ð(u, v) } _|

When the surface is referred to three-dimensional polar coordinates r,
[theta], [phi] given by the equations

x = r sin [theta] cos [phi], y = r sin [theta] sin [phi],
z = r cos [theta],

and the equation of the surface is of the form r = [f]([theta],
[phi]), the area is expressed by the formula
_ _ _ _
/ / | { / ðr \² } / ðr \² |½
| | r | { r² + ( -------- ) } sin² [theta] + ( ------ ) | d[theta] d[phi].
_/_/ |_ { \ð[theta]/ } \ð[phi]/ _|

The surface integral of a function of ([theta], [phi]) over the
surface of a sphere r = const. can be expressed in the form

_ _
/2[pi] /[pi]
| d[phi] | F([theta], [phi]) r² sin [theta] d[theta].
_/ 0 _/ 0

In every case the domain of integration must be chosen so as to
include the whole surface.

(ix.) In three-dimensional polar coordinates the Jacobian

ð(x, y, z)
-------------------- = r² sin [theta]
ð(r, [theta], [phi])

The volume integral of a function F (r, [theta], [phi]) through the
volume of a sphere r = a is
_ _ _
/ a /2[pi] /[pi]
| dr | d[phi] | F(r, [theta], [phi]) r² sin [theta] d[theta].
_/ 0 _/ 0 _/ 0

(x.) Integrations of rational functions through the volume of an
ellipsoid x²/a² + y²/b² + z²/c² = 1 are often effected by means of a
general theorem due to Lejeune Dirichlet (1839), which is as follows:
when the domain of integration is that given by the inequality

/x1\[alpha]1 /x2\^[alpha]2 /x_n\[alpha]_n
( -- ) + ( -- ) + ... + ( --- ) [<=] 1
\a1/ \a2/ \a_n/

where the a's and [alpha]'s are positive, the value of the integral
_ _
/ /
| | ... x1^(n1-1)·x2^(n2-1) ... dx1 dx2 ...
_/_/

a1^(n1) a2^(n2) ... [Gamma] (n1/[alpha]1) [Gamma] (n2/[alpha]2)
is --------------------- ---------------------------------------------.
[alpha]1 [alpha]2 ... [Gamma](1 + n1/[alpha]1 + n2/[alpha]2 + ... )

If, however, the object aimed at is an integration through the volume
of an ellipsoid it is simpler to reduce the domain of integration to
that within a sphere of radius unity by the transformation x = a[xi],
y = b[eta], z = c[zeta], and then to perform the integration through
the sphere by transforming to polar coordinates as in (ix).

Approximate and Mechanical Integration.

56. Methods of approximate integration began to be devised very early.
Kepler's practical measurement of the focal sectors of ellipses (1609)
was an approximate integration, as also was the method for the
quadrature of the hyperbola given by James Gregory in the appendix to
his _Exercitationes geometricae_ (1668). In Newton's _Methodus
differentialis_ (1711) the subject was taken up systematically.
Newton's object was to effect the approximate quadrature of a given
curve by making a curve of the type

y = a0 + a1x + a2x² + ... + a_n x^n

pass through the vertices of (n + 1) equidistant ordinates of the
given curve, and by taking the area of the new curve so determined as
an approximation to the area of the given curve. In 1743 Thomas
Simpson in his _Mathematical Dissertations_ published a very
convenient rule, obtained by taking the vertices of three consecutive
equidistant ordinates to be points on the same parabola. The distance
between the extreme ordinates corresponding to the abscissae x = a and
x = b is divided into 2n equal segments by ordinates y1, y2, ...
y(2n-1), and the extreme ordinates are denoted by y0, y(2n). The
vertices of the ordinates y0, y1, y2 lie on a parabola with its axis
parallel to the axis of y, so do the vertices of the ordinates y2, y3,
y4, and so on. The area is expressed approximately by the formula

{(b - a)/6n} [y0 + y_(2n) + 2 (y2 + y4 + ... + y_(2n-2))
+ 4(y1 + y3 + ... + y_(2n-1)],

which is known as Simpson's rule. Since all simple integrals can be
represented as areas such rules are applicable to approximate
integration in general. For the recent developments reference may be
made to the article by A. Voss in _Ency. d. Math. Wiss._, Bd. II., A.
2 (1899), and to a monograph by B. P. Moors, _Valeur approximative
d'une intégrale définie_ (Paris, 1905).

Many instruments have been devised for registering mechanically the
areas of closed curves and the values of integrals. The best known are
perhaps the "planimeter" of J. Amsler (1854) and the "integraph" of
Abdank-Abakanowicz (1882).

BIBLIOGRAPHY.--For historical questions relating to the subject the
chief authority is M. Cantor, _Geschichte d. Mathematik_ (3 Bde.,
Leipzig, 1894-1901). For particular matters, or special periods, the
following may be mentioned: H. G. Zeuthen, _Geschichte d. Math. im
Altertum u. Mittelalter_ (Copenhagen, 1896) and _Gesch. d. Math. im
XVI. u. XVII. Jahrhundert_ (Leipzig, 1903); S. Horsley, _Isaaci
Newtoni opera quae exstant omnia_ (5 vols., London, 1779-1785); C. I.
Gerhardt, _Leibnizens math. Schriften_ (7 Bde., Leipzig, 1849-1863);
Joh. Bernoulli, _Opera omnia_ (4 Bde., Lausanne and Geneva, 1742).
Other writings of importance in the history of the subject are cited
in the course of the article. A list of some of the more important
treatises on the differential and integral calculus is appended. The
list has no pretensions to completeness; in particular, most of the
recent books in which the subject is presented in an elementary way
for beginners or engineers are omitted.--L. Euler, _Institutiones
calculi differentialis_ (Petrop., 1755) and _Institutiones calculi
integralis_ (3 Bde., Petrop., 1768-1770); J. L. Lagrange, _Leçons sur
le calcul des fonctions_ (Paris, 1806, _Oeuvres_, t. x.), and _Théorie
des fonctions analytiques_ (Paris, 1797, 2nd ed., 1813, _Oeuvres_, t.
ix.); S. F. Lacroix, _Traité de calcul diff. et de calcul int._ (3
tt., Paris, 1808-1819). There have been numerous later editions; a
translation by Herschel, Peacock and Babbage of an abbreviated edition
of Lacroix's treatise was published at Cambridge in 1816. G. Peacock,
_Examples of the Differential and Integral Calculus_ (Cambridge,
1820); A. L. Cauchy, _Résumé des leçons ... sur le calcul
infinitésimale_ (Paris, 1823), and _Leçons sur le calcul différentiel_
(Paris, 1829; _Oeuvres_, sér. 2, t. iv.); F. Minding, _Handbuch d.
Diff.-u. Int.-Rechnung_ (Berlin, 1836); F. Moigno, _Leçons sur le
calcul diff._ (4 tt., Paris, 1840-1861); A. de Morgan, _Diff. and Int.
Calc._ (London, 1842); D. Gregory, _Examples on the Diff. and Int.
Calc._ (2 vols., Cambridge, 1841-1846); I. Todhunter, _Treatise on the
Diff. Calc._ and _Treatise on the Int. Calc._ (London, 1852), numerous
later editions; B. Price, _Treatise on the Infinitesimal Calculus_ (2
vols., Oxford, 1854), numerous later editions; D. Bierens de Haan,
_Tables d'intégrales définies_ (Amsterdam, 1858); M. Stegemann,
_Grundriss d. Diff.- u. Int.-Rechnung_ (2 Bde., Hanover, 1862)
numerous later editions; J. Bertrand, _Traité de calc. diff. et int._
(2 tt., Paris, 1864-1870); J. A. Serret, _Cours de calc. diff. et
int._ (2 tt., Paris, 1868, 2nd ed., 1880, German edition by Harnack,
Leipzig, 1884-1886, later German editions by Bohlmann, 1896, and
Scheffers, 1906, incomplete); B. Williamson, _Treatise on the Diff.
Calc._ (Dublin, 1872), and _Treatise on the Int. Calc._ (Dublin, 1874)
numerous later editions of both; also the article "Infinitesimal
Calculus" in the 9th ed. of the _Ency. Brit._; C. Hermite, _Cours
d'analyse_ (Paris, 1873); O. Schlömilch, _Compendium d. höheren
Analysis_ (2 Bde., Leipzig, 1874) numerous later editions; J. Thomae,
_Einleitung in d. Theorie d. bestimmten Integrale_ (Halle, 1875); R.
Lipschitz, _Lehrbuch d. Analysis_ (2 Bde., Bonn, 1877, 1880); A.
Harnack, _Elemente d. Diff.- u. Int.-Rechnung_ (Leipzig, 1882, Eng.
trans. by Cathcart, London, 1891); M. Pasch, _Einleitung in d. Diff.-
u. Int.-Rechnung_ (Leipzig, 1882); Genocchi and Peano, _Calcolo
differenziale_ (Turin, 1884, German edition by Bohlmann and Schepp,
Leipzig, 1898, 1899); H. Laurent, _Traité d'analyse_ (7 tt., Paris,
1885-1891); J. Edwards, _Elementary Treatise on the Diff. Calc._
(London, 1886), several later editions; A. G. Greenhill, _Diff. and
Int. Calc._ (London, 1886, 2nd ed., 1891); É. Picard, _Traité
d'analyse_ (3 tt., Paris, 1891-1896); O. Stolz, _Grundzüge d. Diff.-
u. Int.-Rechnung_ (3 Bde., Leipzig, 1893-1899); C. Jordan, _Cours
d'analyse_ (3 tt., Paris, 1893-1896); L. Kronecker, _Vorlesungen ü. d.
Theorie d. einfachen u. vielfachen Integrale_ (Leipzig, 1894); J.
Perry, _The Calculus for Engineers_ (London, 1897); H. Lamb, _An
Elementary Course of Infinitesimal Calculus_ (Cambridge, 1897); G. A.
Gibson, _An Elementary Treatise on the Calculus_ (London, 1901); É.
Goursat, _Cours d'analyse mathématique_ (2 tt., Paris, 1902-1905);
C.-J. de la Vallée Poussin, _Cours d'analyse infinitésimale_ (2 tt.,
Louvain and Paris, 1903-1906); A. E. H. Love, _Elements of the Diff.
and Int. Calc._ (Cambridge, 1909); W. H. Young, _The Fundamental
Theorems of the Diff. Calc._ (Cambridge, 1910). A résumé of the
infinitesimal calculus is given in the articles "Diff.- u.
Int-Rechnung" by A. Voss, and "Bestimmte Integrale" by G. Brunel in
_Ency. d. math. Wiss._ (Bde. ii. A. 2, and ii. A. 3, Leipzig, 1899,
1900). Many questions of principle are discussed exhaustively by E. W.
Hobson, _The Theory of Functions of a Real Variable_ (Cambridge,
1907). (A. E. H. L.)

INFINITIVE, a form of the verb, properly a noun with verbal functions, but usually taken as a mood (see GRAMMAR). The Latin grammarians gave it the name of _infinitus_ or _infinitivus modus_, i.e. indefinite, unlimited mood, as not having definite persons or numbers.

INFLEXION (from Lat. _inflectere_, to bend), the action of bending inwards, or turning towards oneself, or the condition of being bent or curved. In optics, the term "inflexion" was used by Newton for what is now known as "diffraction of light" (q.v.). For inflexion in geometry see CURVE. Inflexion when used of the voice, in speaking or singing, indicates a change in tone, pitch or expression. In grammar (q.v.) inflexion indicates the changes which a word undergoes to bring it into correct relations with the other words with which it is used. In English grammar nouns, pronouns, adjectives (in their degrees of comparison), verbs and adverbs are inflected. Some grammarians, however, regard the inflexions of adverbs more as an actual change in word-formation.

INFLUENCE (Late Lat. _influentia_, from _influere_, to flow in), a word whose principal modern meaning is that of power, control or action affecting others, exercised either covertly or without visible means or direct physical agency. It is one of those numerous terms of astrology (q.v.) which have established themselves in current language. From the stars was supposed to flow an ethereal stream which affected the course of events on the earth and the fortunes and characters of men. For the law as to "undue influence" see CONTRACT.

INFLUENZA (syn. "grip," _la grippe_), a term applied to an infectious febrile disorder due to a specific bacillus, characterized specially by catarrh of the respiratory passages and alimentary canal, and occurring mostly as an epidemic. The Italians in the 17th century ascribed it to the influence of the stars, and hence the name "influenza." The French name _grippe_ came into use in 1743, and those of _petite poste_ and _petit courier_ in 1762, while _général_ became another synonym in 1780. Apparently the scourge was common; in 1403 and 1557 the sittings of the Paris law courts had to be suspended through it, and in 1427 sermons had to be abandoned through the coughing and sneezing; in 1510 masses could not be sung. Epidemics occurred in 1580, 1676, 1703, 1732 and 1737, and their cessation was supposed to be connected with earthquakes and volcanic eruptions.

The disease is referred to in the works of the ancient physicians, and accurate descriptions of it have been given by medical writers during the last three centuries. These various accounts agree substantially in their narration of the phenomena and course of the disease, and influenza has in all times been regarded as fulfilling all the conditions of an epidemic in its sudden invasion, and rapid and extensive spread. Among the chief epidemics were those of 1762, 1782, 1787, 1803, 1833, 1837 and 1847. It appeared in fleets at sea away from all communication with land, and to such an extent as to disable them temporarily for service. This happened in 1782 in the case of the squadron of Admiral Richard Kempenfelt (1718-1782), which had to return to England from the coast of France in consequence of influenza attacking his crews.

Like cholera and plague, influenza reappeared in the last quarter of the 19th century, after an interval of many years, in epidemic or rather pandemic form. After the year 1848, in which 7963 deaths were directly attributed to influenza in England and Wales, the disease continued prevalent until 1860, with distinct but minor epidemic exacerbations in 1851, 1855 and 1858; during the next decade the mortality dropped rapidly though not steadily, and the diminution continued down to the year 1889, In which only 55 deaths were ascribed to this cause. It is not clear whether the disease ever disappears wholly, and the deaths registered in 1889 are the lowest recorded in any year since the registrar-general's returns began. Occasionally local outbreaks of illness resembling epidemic influenza have been observed during the period of abeyance, as in Norfolk in 1878 and in Yorkshire in 1887; but whether such outbreaks and the so-called "sporadic" cases are nosologically identical with epidemic influenza is open to doubt. The relation seems rather to be similar to that between Asiatic cholera and "cholera nostras." Individual cases may be indistinguishable, but as a factor in the public health the difference between sporadic and epidemic influenza is as great and unmistakable as that between the two forms of cholera. This fact, which had been forgotten by some since 1847 and never learnt by others, was brought home forcibly to all by the visitation of 1889.

According to the exhaustive report drawn up by Dr H. Franklin Parsons for the Local Government Board, the earliest appearances were observed in May 1889, and three localities are mentioned as affected at the same time, all widely separated from each other--namely, Bokhara in Central Asia, Athabasca in the north-west Territories of Canada and Greenland. About the middle of October it was reported at Tomsk in Siberia, and by the end of the month at St Petersburg. During November Russia became generally affected, and cases were noticed in Paris, Berlin, Vienna, London and Jamaica (?). In December epidemic influenza became established over the whole of Europe, along the Mediterranean, in Egypt and over a large area in the United States. It appeared in several towns in England, beginning with Portsmouth, but did not become generally epidemic until the commencement of the new year. In London the full onset of unmistakable influenza dated from the 1st of January 1890. Everywhere it seems to have exhibited the same explosive character when once fully established. In St Petersburg, out of a government staff of 260 men, 220 were taken ill in one night, the 15th of November. During January 1890 the epidemic reached its height in London, and appeared in a large number of towns throughout the British Islands, though it was less prevalent in the north and north-west than in the south. January witnessed a great extension of the disease in Germany, Holland, Switzerland, Austria-Hungary, Italy, Spain and Portugal; but in Russia, Scandinavia and France it was already declining. The period of greatest activity in Europe was the latter half of December and the earlier half of January, with the change of the year for a central point. Other parts of the world affected in January 1890 were Cape Town, Canada, the United States generally, Algiers, Tunis, Cairo, Corsica, Sardinia, Sicily, Honolulu, Mexico, the West Indies and Montevideo. In February the provincial towns of England were most severely affected, the death-rate rising to 27.4, but in London it fell from 28.1 to 21.2, and for Europe generally the back of the epidemic was broken. At the same time, however, it appeared in Ceylon, Penang, Japan, Hong Kong and India; also in West Africa, attacking Sierra Leone, and Gambia in the middle of the month; and finally in the west, where Newfoundland and Buenos Aires were invaded. In March influenza became widely epidemic in India, particularly in Bengal and Bombay, and made its appearance in Australia and New Zealand. In April and May it was epidemic all over Australasia, in Central America, Brazil, Peru, Arabia and Burma. During the summer and autumn it reached a number of isolated islands, such as Iceland, St Helena, Mauritius and Réunion. Towards the close of the year it was reported from Yunnan in the interior of China, from the Shiré Highlands in Central Africa, Shoa in Abyssinia, and Gilgit in Kashmir. In the course of fifteen months, beginning with its undoubted appearance in Siberia in October 1889, it had traversed the entire globe.

The localities attacked by influenza in 1889-1890 appear in no case to have suffered severely for more than a month or six weeks. Thus in Europe and North America generally the visitation had come to an end in the first quarter of 1890. The earliest signs of an epidemic revival on a large scale occurred in March 1891, in the United States and the north of England. It was reported from Chicago and other large towns in the central states, whence it spread eastwards, reaching New York about the end of March. In England it began in the Yorkshire towns, particularly in Hull, and also independently in South Wales. In London influenza became epidemic for the second time about the end of April, and soon afterwards was widely distributed in England and Wales. The large towns in the north, together with London and Wales, suffered much more heavily in mortality than in the previous attack, but the south-west of England, Scotland and Ireland escaped with comparatively little sickness. The same may be said of the European continent generally, except parts of Russia, Scandinavia and perhaps the north of Germany. This second epidemic coincided with the spring and early summer; it had subsided in London by the end of June. The experience of Sheffield is interesting. In 1890 the attack, contrary to general experience, had been undecided, lingering and mild; in 1891 it was very sudden and extremely severe, the death-rate rising to 73.4 during the month of April, and subsiding with equal rapidity. During the third quarter of the year, while Europe was free, the antipodes had their second attack, which was more severe than the first. As in England, it reversed the previous order of things, beginning in the provinces and spreading thence to the capital towns. The last quarter of the year was signalized by another recrudescence in Europe, which reached its height during the winter. All parts, including Great Britain, were severely affected. In England those parts which had borne the brunt of the epidemic in the early part of the year escaped. In fact, these two revivals may be regarded as one, temporarily interrupted by the summer quarter.

The recrudescence at the end of 1891 lasted through mid-winter, and in many places, notably in London, it only reached its height in January 1892, subsiding slowly and irregularly in February and March. Brighton suffered with exceptional severity. The continent of Europe seems to have been similarly affected. In Italy the notifications of influenza were as follow: 1891--January to October, 0; November, 30; December, 6461; 1892--January, 84,543; February, 55,352; March, 28,046; April, 7962; May, 1468; June, 223. Other parts of the world affected were the West Indies, Tunis, Egypt, Sudan, Cape Town Teheran, Tongking and China. In August 1892 influenza was reported from Peru, and later in the year from various places in Europe.

A fourth recrudescence, but of a milder character, occurred in Great Britain in the spring of 1893, and a fifth in the following winter, but the year 1894 was freer from influenza than any since 1890. In 1895 another extensive epidemic took place. In 1896 influenza seemed to have spent its strength, but there was an increased prevalence of the disease in 1897, which was repeated on a larger scale in 1898, and again in 1899, when 12,417 deaths were recorded in England and Wales. This was the highest death-rate since 1892. After this the death-rate declined to half that amount and remained there with the slight upward variations until 1907, in which the total death-rate was 9257. The experience of other countries has been very similar; they have all been subjected to periodical revivals of epidemic influenza at irregular intervals and of varying intensity since its reappearance in 1889, but there has been a general though not a steady decline in its activity and potency. Its behaviour is, in short, quite in keeping with the experience of 1847-1860, though the later visitation appears to have been more violent and more fatal than the former. Its diffusion was also more rapid and probably more extensive.

The foregoing general summary may be supplemented by some further details of the incidence in Great Britain. The number of deaths directly attributed to influenza, and the death-rates per million in each year in England and Wales, are as follow:--

+------+--------+-------------+
| Year.| Deaths.| Death-rates |
| | | per million.|
+------+--------+-------------+
| 1890 | 4,523 | 157 |
| 1891 | 16,686 | 574 |
| 1892 | 15,737 | 534 |
| 1893 | 9,669 | 325 |
| 1894 | 6,625 | 220 |
| 1895 | 12,880 | 424 |
| 1896 | 3,753 | 122 |
| 1897 | 6,088 | 196 |
| 1898 | 10,405 | 331 |
| 1899 | 12,417 | 389 |
| 1900 | 16,245 | 504 |
| 1901 | 5,666 | 174 |
| 1902 | 7,366 | 223 |
| 1903 | 6,322 | 189 |
| 1904 | 5,694 | 168 |
| 1905 | 6,953 | 204 |
| 1906 | 6,310 | 183 |
| 1907 | 9,257 | 265 |
+------+--------+-------------+

It is interesting to compare these figures with the corresponding ones for the previous visitation:--

+------+--------+-------------+
| Year.| Deaths.| Death-rates |
| | | per million.|
+------+--------+-------------+
| 1847 | 4,881 | 285 |
| 1848 | 7,963 | 460 |
| 1849 | 1,611 | 92 |
| 1850 | 1,380 | 78 |
| 1851 | 2,152 | 120 |
| 1852 | 1,359 | 76 |
| 1853 | 1,789 | 99 |
| 1854 | 1,061 | 58 |
| 1855 | 3,568 | 193 |
+------+--------+-------------+

The two sets of figures are not strictly comparable, because, during the first period, notification of the cause of death was not compulsory; but it seems clear that the later wave was much the more deadly. The average annual death-rate for the nine years is 320 in the one case against 162 in the other, or as nearly as possible double. In both epidemic periods the second year was far more fatal than the first, and in both a marked revival took place in the ninth year; in both also an intermediate recrudescence occurred, in the fifth year in one case, in the sixth in the other. The chief point of difference is the sudden and marked drop in 1849-1850, against a persistent high mortality in 1892-1893, especially in 1892, which was nearly as fatal as 1891.

To make the significance of these epidemic figures clear, it should be added that in the intervening period 1861-1889 the average annual death-rate from influenza was only fifteen, and in the ten years immediately preceding the 1890 outbreak it was only three. Moreover, in epidemic influenza, the mortality directly attributed to that disease is only a fraction of that actually caused by it. For instance, in January 1890 the deaths from influenza in London were 304, while the excess of deaths from respiratory diseases was 1454 and from all causes 1958 above the average.

We have seen above that the mortality was far greater in the second epidemic year than in the first, and this applies to all parts of England, and to rural as well as to urban communities, as the following table shows:--

_Deaths from Influenza._

+--------------------------------------+------+------+
| | 1890.| 1891.|
+--------------------------------------+------+------+
| London | 624 | 2302 |
| 24 Great Towns over 80,000 population| 439 | 2417 |
| 35 Towns between 20,000 and 80,000 | 186 | 765 |
| 21 Towns between 10,000 and 20,000 | 46 | 196 |
| 60 Towns under 10,000 | 62 | 196 |
| 85 Rural Sanitary Districts | 317 | 841 |
+--------------------------------------+------+------+

In spite of these figures, it appears that the 1890 attack, which was in general much more sudden in its onset than that of 1891, also caused a great deal more sickness. More people were "down with influenza," though fewer died. For Instance, the number of persons treated at the Middlesex Hospital in the two months' winter epidemic of 1890 was 1279; in the far more fatal three months' spring epidemic of 1891 it was only 726. One explanation of this discrepancy between the incidence of sickness and mortality is that in the second attack, which was more protracted and more insidious, the stress of the disease fell more upon the lungs. Another is that its comparative mildness, combined with the time of year, in itself proved dangerous, because it tempted people to disregard the illness, whereas in the first epidemic they were too ill to resist. On the whole, rural districts showed a higher death-rate than towns, and small towns a higher one than large ones in both years. This is explained by the age distribution in such localities; influenza being particularly fatal to aged people, though no age is exempt. Certain counties were much more severely affected than others. The eastern counties, namely, Essex, Suffolk and Norfolk, together with Hampshire and one or two others, escaped lightly in both years; the western counties, namely, North and South Wales, with the adjoining counties of Monmouth, Hereford and Shropshire, suffered heavily in both years.

It will be convenient to discuss _seriatim_ the various points of interest on which light has been thrown by the experience described above.

The bacteriology of influenza is discussed in the article on PARASITIC DISEASES. The disease is often called "Russian" influenza, and its origin in 1889 suggests that the name may have some foundation in fact. A writer, who saw the epidemic break out in Bokhara, is quoted by him to the following effect:--"The summer of 1888 was exceptionally hot and dry, and was followed by a bitterly cold winter and a rainy spring. The dried-up earth was full of cracks and holes from drought and subsequent frost, so that the spring rains formed ponds in these holes, inundated the new railway cuttings, and turned the country into a perfect marsh. When the hot weather set in the water gave off poisonous exhalations, rendering malaria general." On account of the severe winter, the people were enfeebled from lack of nourishment, and when influenza broke out suddenly they died in large numbers. Europeans were very severely affected. Russians, hurrying home, carried the disease westwards, and caravans passing eastwards took it into Siberia. There is a striking similarity in the conditions described to those observed in connexion with outbreaks of other diseases, particularly typhoid fever and diphtheria, which have occurred on the supervention of heavy rain after a dry period, causing cracks and fissures in the earth. Assuming the existence of a living poison in the ground, we can easily understand that under certain conditions, such as an exceptionally dry season, it may develop exceptional properties and then be driven out by the subsequent rains, causing a violent outbreak of illness. Some such explanation is required to account for the periodical occurrence of epidemic and pandemic diffusions starting from an endemic centre. We may suppose that a micro-organism of peculiar robustness and virulence is bred and brought into activity by a combination of favourable conditions, and is then disseminated more or less widely according to its "staying power," by human agency. Whether central Asia is an endemic centre for influenza or not there is no evidence, but the disease seems to be more often prevalent in the Russian Empire than elsewhere. Extensive outbreaks occurred there in 1886 and 1887, and it is certain that the 1889 wave was active in Siberia at an earlier date than in Europe, and that it moved eastwards. The hypothesis that it originated in China is unsupported by evidence. But whatever may be the truth with regard to origin, the dissemination of influenza by human agency must be held to be proved. This is the most important addition to our knowledge of the subject contributed by recent research. The upshot of the inquiry by Dr Parsons was to negative all theories of atmospheric influence, and to establish the conclusion that the disease was "propagated mainly, perhaps entirely, by human intercourse."

He found that it prevailed independently of climate, season and
weather; that it moved in a contrary direction to the prevailing
winds; that it travelled along the lines of human intercourse, and not
faster than human beings can travel; that in 1889 it travelled much
faster than in previous epidemics, when the means of locomotion were
very inferior; that it appeared first in capital towns, seaports and
frontier towns, and only affected country districts later; that it
never commenced suddenly with a large number of cases in a place
previously free from disease, but that epidemic manifestations were
generally preceded for some days or weeks by scattered cases; that
conveyance of infection by individuals and its introduction into fresh
places had been observed in many instances; that persons brought much
into contact with others were generally the first to suffer; that
persons brought together in large numbers in enclosed spaces suffered
more in proportion than others, and that the rapidity and extent of
the outbreak in institutions corresponded with the massing together of
the inmates.

These conclusions, based upon the 1889-1890 epidemic, have been confirmed by subsequent experience, especially in regard to the complete independence of season and weather shown by influenza. It has appeared and disappeared at all seasons and in all weathers and only popular ignorance continues to ascribe its behaviour to atmospheric conditions. In Europe, however, it has prevailed more often in winter than in summer, which may be due to the greater susceptibility of persons in winter, or, more probably, to the fact that they congregate more in buildings and are less in the open air during that part of the year. No doubt is any longer entertained of its infectious character, though the degree of infectivity appears to vary considerably. Many cases have been recorded of individuals introducing it into houses, and of all or most of the other inmates then taking it from the first case. Difficulties in preventing the spread of infection are due to (1) the shortness of the period of incubation, (2) the disease being infectious in the earliest stages before the nature of the illness is recognized, (3) the milder varieties being equally infectious with the severe attacks, and the patient going to work and spreading the infection, (4) the diagnosis often being difficult, influenza being possibly confused with ordinary catarrhal attacks, typhoid fever and other diseases. Domestic animals seem to be free from any suspicion of being liable to human influenza. Sanitary conditions, other than overcrowding, do not appear to exercise any influence on the spread of influenza.

Influenza has been shown to be an acute specific fever having nothing whatever to do with a "bad cold." There may be some inflammation of the respiratory passages, and then symptoms of catarrh are present, but that is not necessarily the case, and in some epidemics such symptoms are quite exceptional. This had been recognized by various writers before the 1889 visitation, but it had not been generally realized, as it has been since, and some medical authorities, who persisted in regarding influenza as essentially a "catarrhal" affection, were chiefly to blame for a widespread and tenacious popular fallacy.

Leichtenstern, in his masterly article in Nothnagel's _Handbuch_, divides the disease as follows:--(1) Epidemic influenza vera caused by Pfeiffer's bacillus; (2) Endemic-epidemic influenza vera, which occurs several years after a pandemic and is caused by the same bacillus; (3) Endemic influenza nostras or eatarrhal fever, called _la grippe_, and bearing the same relation to true influenza as cholera nostras does to Asiatic cholera.

The "period of incubation" is one to four days. Susceptibility varies greatly, but the conditions that influence it are matters of conjecture only. It appears that the inhabitants of Great Britain are less susceptible than those of many other countries. Dr Parsons gives the following list, showing the proportion of the population estimated to have been attacked in the 1889-1890 epidemic in different localities:--

+---------------------+---------+---------------------+---------+
| Place. |Per cent.| Place. |Per cent.|
+---------------------+---------+---------------------+---------+
| St Petersburg | 50 | Portugal | 90 |
| Berlin | 33 | Vienna | 30-40 |
| Nuremberg | 67 | Belgrade | 33 |
| Grand-Duchy of Hesse| 25-30 | Antwerp | 33 |
| Grand-Duchy, other | | Gaeta | 50-77 |
| Districts | 50-75 | Massachusetts | 39 |
| Heligoland | 50 | Peking | 50 |
| Budapest | 50 | St Louis (Mauritius)| 67 |
+---------------------+---------+---------------------+---------+

In and about London he reckoned roughly from a number of returns that the proportion was about 12½% among those employed out of doors and 25% among those in offices, &c. The proportion among the troops in the Home District was 9.3%. The General Post Office made the highest return with 33.6%, which is accounted for partly by the enormous number of persons massed together in the same room in more than one department, and partly by the facilities for obtaining medical advice, which would tend to bring very light cases, unnoticed elsewhere, upon the record. No public service was seriously disorganized in England by sickness in the same manner as on the continent of Europe. Some individuals appear to be totally immune; others take the disease over and over again, deriving no immunity, but apparently greater susceptibility from previous attacks.

The symptoms were thus described by Dr Bruce Low from observations made in St Thomas's Hospital, London, in January 1890:--

The invasion is sudden; the patients can generally tell the time when
they developed the disease; e.g. acute pains in the back and loins
came on quite suddenly while they were at work or walking in the
street, or in the case of a medical student, while playing cards,
rendering him unable to continue the game. A workman wheeling a barrow
had to put it down and leave it; and an omnibus driver was unable to
pull up his horses. This sudden onset is often accompanied by vertigo
and nausea, and sometimes actual vomiting of bilious matter. There are
pains in the limbs and general sense of aching all over; frontal
headache of special severity; pains in the eyeballs, increased by the
slightest movement of the eyes; shivering; general feeling of misery
and weakness, and great depression of spirits, many patients, both men
and women, giving way to weeping; nervous restlessness; inability to
sleep, and occasionally delirium. In some cases catarrhal symptoms
develop, such as running at the eyes, which are sometimes injected on
the second day; sneezing and sore throat; and epistaxis, swelling of
the parotid and submaxillary glands, tonsilitis, and spitting of
bright blood from the pharynx may occur. There is a hard, dry cough of
a paroxysmal kind, worst at night. There is often tenderness of the
spleen, which is almost always found enlarged, and this persists after
the acute symptoms have passed. The temperature is high at the onset
of the disease. In the first twenty-four hours its range is from 100°
F. in mild cases to 105° in severe cases.

Dr J. S. Bristowe gave the following description of the illness during the same epidemic:--

The chief symptoms of influenza are, coldness along the back, with
shivering, which may continue off and on for two or three days; severe
pain in the head and eyes, often with tenderness in the eyes and pain
in moving them; pains in the ears; pains in the small of the back;
pains in the limbs, for the most part in the fleshy portions, but also
in the bones and joints, and even in the fingers and toes; and febrile
temperature, which may in the early period rise to 104° or 105° F. At
the same time the patient feels excessively ill and prostrate, is apt
to suffer from nausea or sickness and diarrhoea, and is for the most
part restless, though often (and especially in the case of children
and those advanced in age) drowsy.... In ordinary mild cases the above
symptoms are the only important ones which present themselves, and the
patient may recover in the course of three or four days. He may even
have it so mildly that, although feeling very ill, he is able to go
about his ordinary work. In some cases the patients have additionally
some dryness or soreness of the throat, or some stiffness and
discharge from the nose, which may be accompanied by slight bleeding.
And in some cases, for the most part in the course of a few days, and
at a time when the patient seems to be convalescent, he begins to
suffer from wheezing in the chest, cough, and perhaps a little
shortness of breath, and before long spits mucus in which are
contained pellets streaked or tinged with blood.... Another
complication is diarrhoea. Another is a roseolous spotty rash....
Influenza is by no means necessarily attended with the catarrhal
symptoms which the general public have been taught to regard as its
distinctive signs, and in a very large proportion of cases no
catarrhal condition whatever becomes developed at any time.

Several writers have distinguished four main varieties of the disease--namely, (1) nervous, (2)gastro-intestinal, (3)respiratory, (4) febrile, a form chiefly found in children. Clifford Allbutt says, "Influenza simulates other diseases." Many forms are of typhoid or comatose types. Cardiac attacks are common, not from organic disease but from the direct poisoning of the heart muscle by influenza.